Let's simplify and evaluate the given expression step by step.

The expression is:

$\frac{2(3x - 1.25) \sin^3(\pi x)}{3} + \left(\frac{9 \left(\frac{\sin(2\pi x)}{2\pi x} - x \right)}{10} \times \left(\sin(\pi x) + \sin(\pi(1.25 - 2x)) - 0.0443 \pi (3x - 1.25) \right)\right)$

Step-by-Step Simplification:

1. Simplify the first term: $\frac{2(3x - 1.25) \sin^3(\pi x)}{3}$

2. Simplify the second term inside the parentheses: $\frac{9 \left(\frac{\sin(2\pi x)}{2\pi x} - x \right)}{10}$

3. Combine and simplify the expressions within the second term: $\sin(\pi x) + \sin(\pi(1.25 - 2x)) - 0.0443 \pi (3x - 1.25)$

Detailed Simplification:

1. First term: $\frac{2(3x - 1.25) \sin^3(\pi x)}{3}$

2. Second term: $\frac{9 \left(\frac{\sin(2\pi x)}{2\pi x} - x \right)}{10} \times \left(\sin(\pi x) + \sin(\pi(1.25 - 2x)) - 0.0443 \pi (3x - 1.25)\right)$

To get a numerical or further simplified expression, we would need the value of $x$. For now, we can only simplify algebraically as follows:

• Simplify the coefficient and sine terms separately.

Combining them, we get the final simplified algebraic expression: $\frac{2(3x - 1.25) \sin^3(\pi x)}{3} + \left(\frac{9 \left(\frac{\sin(2\pi x)}{2\pi x} - x \right)}{10} \times \left(\sin(\pi x) + \sin(\pi(1.25 - 2x)) - 0.0443 \pi (3x - 1.25)\right)\right)$

If you provide a specific value for $x$, we can further evaluate this expression numerically.

Would you like more details on any particular step, or have any questions about this?

Here are some follow-up questions you might ask:

1. How do you simplify trigonometric expressions?
2. What are some common trigonometric identities used in simplifications?
3. Can you provide examples of simplifying similar expressions with specific values?
4. How do you deal with composite functions in trigonometric equations?
5. What are the properties of the sine function that are useful in solving such problems?

Tip: Always check if a trigonometric expression can be factored or if identities can be applied to simplify the problem before proceeding with detailed calculations.